地點河工二館匯總課件

1、Null-field approach for Laplace problems with circular boundaries using degenerate kernels沈文成 陳正宗 時間：13:20 14:00地點：河工二館 307 室BEM course May 13, 2008 (typicalBVP-L.ppt)1Null-field approach for LaplacOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution an

2、d boundary density Adaptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions2OutlinesMotivation and literatOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Ada

3、ptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions3OutlinesMotivation and literatMotivation and literature reviewFictitious BEMBEM/BIEMNull-field approachBump contourLimit process Singular and hypersingularRegularImproper integ

4、ralCPV and HPVIll-posedFictitious boundaryCollocation point4Motivation and literature reviPresent approach1. No principal value2. Well-posedAdvantages of degenerate kernelDegenerate kernelFundamental solutionCPV and HPVNo principal value5Present approach1. No principEngineering problem with arbitrar

5、y geometriesDegenerate boundaryCircular boundaryStraight boundaryElliptic boundary(Fourier series)(Legendre polynomial)(Chebyshev polynomial)(Mathieu function)6Engineering problem with arbitMotivation and literature reviewAnalytical methods for solving Laplace problems with circular holesConformal m

6、appingBipolar coordinateSpecial solutionLimited to doubly connected domainLebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover PublicationsChen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied MechanicsHonein, Honein a

7、nd Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics7Motivation and literature reviFourier series approximationLing (1943) - torsion of a circular tubeCaulk et al. (1983) - steady heat conduction with circular holesBird and Steele (1992) - harmonic and

8、 biharmonic problems with circular holesMogilevskaya et al. (2002) - elasticity problems with circular boundaries8Fourier series approximationLiContribution and goalHowever, they didnt employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although the

9、y all employed Fourier series expansion.To develop a systematic approach for solving Laplace problems with multiple holes is our goal.9Contribution and goalHowever, OutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive obser

10、ver system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions10OutlinesMotivation and literatBoundary integral equation and null-field integral equationInterior caseExterior caseNull-field integral equation11Boundary integral equation andOutlinesMot

11、ivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions12OutlinesMotivation and literatExpansions of fundamental soluti

12、on and boundary densityDegenerate kernel - fundamental solutionFourier series expansions - boundary density13Expansions of fundamental soluSeparable form of fundamental solution (1D)Separable propertycontinuousdiscontinuous14Separable form of fundamental Separable form of fundamental solution (2D)15

13、Separable form of fundamental Boundary density discretizationFourier seriesEx . constant elementPresent methodConventional BEM16Boundary density discretizatioOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive observer sys

14、tem Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions17OutlinesMotivation and literatAdaptive observer systemcollocation point18Adaptive observer systemcollocOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental

15、 solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions19OutlinesMotivation and literatVector decomposition technique for potential gradientSpecial case (concentric case) :Non-concentric case:True n

16、ormal direction20Vector decomposition techniqueOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions21Outl

17、inesMotivation and literatLinear algebraic equationwhereColumn vector of Fourier coefficients(Nth routing circle)Index of collocation circleIndex of routing circle 22Linear algebraic equationwhereExplicit form of each submatrix Upk and vector tkFourier coefficientsTruncated terms of Fourier seriesNu

18、mber of collocation points23Explicit form of each submatriFlowchart of present methodPotential of domain pointAnalyticalNumericalAdaptive observer systemDegenerate kernelFourier seriesLinear algebraic equation Collocation point and matching B.C.Fourier coefficientsVector decompositionPotential gradi

19、ent24Flowchart of present methodPotComparisons of conventional BEM and present methodBoundarydensitydiscretizationAuxiliarysystemFormulationObserversystemSingularityConventionalBEMConstant,Linear,QurdratureFundamentalsolutionBoundaryintegralequationFixedobserversystemCPV, RPVand HPVPresentmethodFour

20、ierseriesexpansionDegeneratekernelNull-fieldintegralequationAdaptiveobserversystemNoprincipalvalue25Comparisons of conventional BEOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition tec

21、hnique Linear algebraic equationNumerical examplesDegenerate scaleConclusions26OutlinesMotivation and literatNumerical examplesSteady state heat conduction problemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shear

22、Half-plane problems27Numerical examplesSteady stateNumerical examplesSteady state heat conduction problemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shearHalf-plane problems28Numerical examplesSteady stateSteady

23、state heat conduction problemsCase 1Case 229Steady state heat conduction pSteady state heat conduction problemsCase 3Case 430Steady state heat conduction pCase 1: Isothermal lineExact solution(Carrier and Pearson)BEM-BEPO2D(N=21)FEM-ABAQUS(1854 elements)Present method(M=10)31Case 1: Isothermal lineE

24、xact sRelative error of flux on the small circle32Relative error of flux on the Convergence test - Parsevals sum for Fourier coefficientsParsevals sum33Convergence test - Parsevals Case 2: Isothermal lineCaulks data (1983)IMA Journal of Applied MathematicsPresent method (M=10)FEM-ABAQUS(6502 element

25、s)34Case 2: Isothermal lineCaulksCase 3: Isothermal lineFEM-ABAQUS(8050 elements)Present method (M=10)Caulks data (1983)IMA Journal of Applied Mathematics35Case 3: Isothermal lineFEM-ABACase 4: Isothermal lineFEM-ABAQUS(8050 elements)Present method (M=10)Caulks data (1983)IMA Journal of Applied Math

26、ematics36Case 4: Isothermal lineFEM-ABANumerical examplesSteady state heat conduction problemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shearHalf-plane problems37Numerical examplesSteady stateElectrostatic poten

27、tial of wiresHexagonal electrostatic potentialTwo parallel cylinders held positive and negative potentials38Electrostatic potential of wirContour plot of potentialExact solution (Lebedev et al.)Present method (M=10)39Contour plot of potentialExactContour plot of potentialOnishis data (1991)Present m

28、ethod (M=10)40Contour plot of potentialOnishNumerical examplesSteady state heat conduction problemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shearHalf-plane problems41Numerical examplesSteady stateFlow of an ide

29、al fluid pass two parallel cylinders is the velocity of flow far from the cylinders is the incident angle42Flow of an ideal fluid pass twVelocity field in different incident anglePresent method (M=10)Present method (M=10)43Velocity field in different iNumerical examplesSteady state heat conduction p

30、roblemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shearHalf-plane problems44Numerical examplesSteady stateTorsion bar with circular holes removedThe warping functionBoundary condition whereonTorque45Torsion bar w

31、ith circular holeAxial displacement with two circular holesPresent method (M=10)Caulks data (1983)ASME Journal of Applied MechanicsDashed line: exact solutionSolid line: first-order solution46Axial displacement with two ciAxial displacement with three circular holesPresent method (M=10)Caulks data (

32、1983)ASME Journal of Applied MechanicsDashed line: exact solutionSolid line: first-order solution47Axial displacement with three Axial displacement with four circular holesPresent method (M=10)Caulks data (1983)ASME Journal of Applied MechanicsDashed line: exact solutionSolid line: first-order solut

33、ion48Axial displacement with four cNumerical examplesSteady state heat conduction problemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shearHalf-plane problems49Numerical examplesSteady stateInfinite medium under a

34、ntiplane shearThe displacementBoundary conditionTotal displacementon50Infinite medium under antiplanShear stress zq around the hole of radius a1 (x axis)Present method (M=20)Honeins data (1992)Quarterly of Applied Mathematics51Shear stress zq around the hoShear stress zq around the hole of radius a1

35、 (y axis)Present method (M=20)Honeins data (1992)Quarterly of Applied Mathematics52Shear stress zq around the hoShear stress zq around the hole of radius a1 (45 degrees)Present method (M=20)Honeins data (1992)Quarterly of Applied Mathematics53Shear stress zq around the hoShear stress zq around the h

36、ole of radius a1 (Touching)Present methoddiscontinuousdiscontinuousHoneins data (1992)Quarterly of Applied MathematicsGibbs phenomenon54Shear stress zq around the hoTwo equivalent approachesDisplacement approachStress approachPresent methodBird and Steele (1992)ASME Journal of Applied Mechanics55Two

37、 equivalent approachesDisplShear stress zq around the hole of radius a1Present method (M=20)Present method (M=20)Steeles data (1992)Stress approachDisplacement approachHoneins data (1992)5.3485.3494.6475.34513.13%0.02%Analytical0.06%56Shear stress zq around the hoConvergence of stress zq at q=45 deg

38、rees versus R057Convergence of stress zq atThree circular holes with centers on the x axis58Three circular holes with centThree circular holes with centers on the y axis59Three circular holes with centThree circular holes with centers on the line making 45 degrees60Three circular holes with centNume

39、rical examplesSteady state heat conduction problemsElectrostatic potential of wiresFlow of an ideal fluid pass cylindersA circular bar under torqueAn infinite medium under antiplane shearHalf-plane problems61Numerical examplesSteady stateHalf-plane problemsDirichlet boundary condition(Lebedev et al.

40、)Mixed-type boundary condition(Lebedev et al.)62Half-plane problemsDirichlet bDirichlet problemExact solution (Lebedev et al.)Present method (M=10)Isothermal line63Dirichlet problemExact solutioMixed-type problemExact solution (Lebedev et al.)Present method (M=10)Isothermal line64Mixed-type problemE

41、xact solutiOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions65OutlinesMotivation and literatNumerical

42、instability in BEMAnnular caseInterior caseMax errorDegenerate scaleInternational Journal forNumerical Methods in EngineeringEngineering Analysiswith Boundary Elements Matrix singularErrorSingularvalue66Numerical instability in BEMAnDegenerate scale in the multiply connected problema1 =1.0, influenc

43、e matrix U is singular67Degenerate scale in the multipTreatments of degenerate scale problemMethod of adding a rigid body termCHEEF conceptSingularSingularAuxiliary constraint NonsingularCHEEF pointPromote rank68Treatments of degenerate scaleThe minimum singular value versus radius a1Degenerate scal

44、eNumerical failure69The minimum singular value verOutlinesMotivation and literature reviewMathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equationNumerical examplesDegenerate scaleConclusions70O

45、utlinesMotivation and literatConclusionsA systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace problems with circular boundaries.Numerical results agree well with available exact solutions, Caulks data, Onishis data and FEM (ABAQUS) for only few terms of Fourier series.71ConclusionsA systematic approaConclusionsMethod of adding a rigid body term and CHEEF approach have been successfully adopted to overcome the degenerate scale for multiply